Combine shredded chicken, buffalo sauce, melted butter, and cheese in a mixing bowl, stirring until the mixture is rich, glossy, and fully coated.
Warm each tortilla in a dry skillet over medium heat for 15–20 seconds per side until soft and flexible to prevent cracking.
Add about 2 tablespoons of filling to each tortilla, roll tightly, and place seam-side down on a baking sheet to hold their shape.
Heat oil in a skillet over medium heat until shimmering, then fry the taquitos seam-side down for 1–2 minutes per side until golden and crisp. Transfer to paper towels to drain.
Mix sour cream (or Greek yogurt) with ranch or blue cheese dressing and a little chopped chives in a small bowl to make a cool, creamy dip.
Arrange the taquitos on a platter, drizzle with extra buffalo sauce, and serve with the dip. Add more chives or crumbled cheese on top if you like.